All elements of a group have a unique inverse

up:: Group

Let $G$ be a group. Have $a\in G$ be an element, and suppose that it has two inverses $a^{-1},a^{\prime }\in G$. Then we have that

$$e=a{a}^{-1}=a{a}^{\prime }$$

Multiplying on the left by $a^{-1}$ (or by $a^{\prime }$, whichever), yields

$$a^{-1}=a^{\prime }$$